TIPE

G= (V, E)p∈[0,1]
G0= (V, E0)E E0
p
G0
Ω = Qe∈E{0,1}Ω
EF
µp
{0,1}µp({1}) = p µp({0})=1−p Pp
Pp=Qe∈Eµp(Ω,F, Pp)
E E0
x∈V CxE0x
G
GL2V=Z2E{x, y}
x, y ∈Z2kx−yk= 1
Ld
ψ(p)θ(p)
0
x x
θ(p)
A
Pp(A)∈ {0,1}.
B
Pp(A|B) = Pp(A)Pp(A)>0Pp(B|A) = Pp(B)Pp(A|B)
Pp(A)=Pp(B)
PpE→Pp(E|A) (Ω,F)
E
F0E∈FPp(E) = Pp(E|A)
F E F 0=F
Pp(A) = Pp(A|A)=1
G0
(xn) (xn, xn+1)nN
G0N(xn, xn+1)
n≥N
θ pc(L2) = sup{p∈[0,1], θ(p) = 0}
pcθ(1) = 1 >0
p>pcψ(p)≥θ(p)>0ψ(p) = 1
p<pcθ(p)=0 θ x
xZ2
ψ(p)=0

ψ(p) = 0 p<pcψ(p)=1 p>pc
pc
0< pc<1
0< pc<1
nN∗P(n)
0
P(n)
θ(p)≤Pp(c∈P(n) )
≤X
c∈P(n)
Pp(c)
=|P(n)|pn
≤4npn
p < 1
4θ(p)=0 pc≥1 4>0
pc<1L∗L2
x+ (1
2,1 2)x∈Z2{x, y} kx−yk= 1
eL2e∗L∗
L2
e∗e
L2

pc>0
0γ0
nN∗Γ(n)
n x =1
2
(1
2, k +1 2) 0 ≤k≤n|Γ(n)| ≤ (n+ 1)4n
1−θ(p)≤X
γ
Pp(γ)
≤X
n∈N∗|Γ(n)|(1 −p)n
≤X
n∈N∗
(n+ 1)4n(1 −p)n
1−p
1−θ(p)≤1
21−p
k.k k(x, y)k=|x|+|y|x
y A r(A) = maxx∈Akxk
S(x, n)x n S(n) = S(0, n)
p < pcξ(p)>0n
N
Pp(r(C0)≥n)≤e−ξ(p)n
Pp(r(C0)≥n)
pc
pc≤1
2

ω≤ω0ω, ω0∈Ωω(e)≤ω0(e)
e
A∈Fω∈A ω0≥ω
ω0∈A
A B A$B
α β A
α B
β
A$B A B
A B
Pp(A$B)≤Pp(A)Pp(B)
A ω ∈Ωe
A ω ω ∈A ω0ω e
ω /∈A
A N(A)
A
A
d
dpPp(A) = Ep(N(A))
(Xn)
N
N
E(X1+... +XN) = E(N)E(X1)
p∈]0, pc[n∈N∗Anr(C0)≥n
gn(p) = Pp(An)
g0
n(p) = 1
pEp(N(An)|An)gn(p)
Ep(N(An)|An)
An0x
kxk=n e1, ..., eN(An)